Problem 54 - Algebra and Trigonometry

Question

When Are Both Products Defined? What must be true about the dimensions of the matrices \(A\) and \(B\) if both products \(A B\) and \(B A\) are defined?

Step-by-Step Solution

Answer

Matrices are compatible for both products if \(A: m \times n\) and \(B: n \times m\).

Step 1: Understand Matrix Multiplication

In matrix multiplication, two matrices can be multiplied if the number of columns in the first matrix matches the number of rows in the second matrix.

Step 2: Condition for AB to be Defined

For the product \(AB\) to be defined, the number of columns in matrix \(A\) must be equal to the number of rows in matrix \(B\). If matrix \(A\) is of dimensions \(m \times n\), then matrix \(B\) must have dimensions \(n \times p\) (where \(p\) can be any number).

Step 3: Condition for BA to be Defined

For the product \(BA\) to be defined, the number of columns in matrix \(B\) must be equal to the number of rows in matrix \(A\). Referring to the previous step, matrix \(B\) is \(n \times p\), and matrix \(A\) is \(m \times n\). Therefore, it must also be true that \(p = m\).

Step 4: Conclusion about Matrix Dimensions

Based on Steps 2 and 3, both products \(AB\) and \(BA\) are defined only if matrix \(A\) is \(m \times n\), and matrix \(B\) is \(n \times m\). That is, the number of columns in \(A\) must equal the number of rows in \(B\), and vice versa.

Key Concepts

Matrix Dimensions

Matrix dimensions refer to the size of the matrix in terms of rows and columns. When you talk about the dimensions of a matrix, you express it as "number of rows" by "number of columns", or in mathematical terms, an \( m \times n \) matrix. Here, \( m \) is the number of rows and \( n \) is the number of columns.

Understanding matrix dimensions is crucial when you're dealing with matrix operations like multiplication. Before you can even attempt to multiply two matrices, you need to know their dimensions. This will determine whether the multiplication is actually possible.

Why are matrix dimensions important?

• They dictate the structure and shape of a matrix.
• They determine whether two matrices can be multiplied.
• They affect the size of the resulting matrix product.

So, always check the dimensions before proceeding with matrix calculations.

Matrix Product

The matrix product is the result you get when you multiply two matrices together. But matrix multiplication is not as straightforward as multiplying numbers.

In matrix multiplication, things work a bit differently from typical arithmetic multiplication. You can't just multiply any two matrices together; they must satisfy certain conditions related to their dimensions, which we'll discuss next.

Here’s a quick overview of what happens during matrix multiplication:

• The elements of the resulting matrix are obtained by performing dot products between the rows of the first matrix and the columns of the second matrix.
• The dimension of the resulting product is determined by the number of rows of the first matrix and the number of columns of the second matrix.
• Each element in the product is calculated by summing the products of elements from corresponding positions in the row of the first and the column of the second matrix.

This might sound complex, but with practice, it becomes much clearer.

Multiplication Condition

The multiplication condition is essential for determining when matrix multiplication can actually take place. To multiply two matrices, there is an important rule about their dimensions that must be met.

For two matrices to be multiplied:

• The number of columns in the first matrix must equal the number of rows in the second matrix.

So if you have a matrix \( A \) with dimensions \( m \times n \), you can only multiply it with a matrix \( B \) with dimensions \( n \times p \) (where \( p \) can be any number). The resulting matrix product will then have dimensions \( m \times p \).

Remember:

• If this condition is not met, the multiplication is not defined.
• It's also why matrix multiplication is not commutative, i.e., \( AB eq BA \) in general.

This makes understanding the multiplication condition critical when working with matrices.

AB and BA Defined

Sometimes, you need to know not only that you can multiply two matrices but that you can do so in either order. This happens rarely, but it does come up.

For both \( AB \) and \( BA \) to be defined:

• The number of columns in matrix \( A \) must equal the number of rows in matrix \( B \).
• Additionally, the number of columns in matrix \( B \) must equal the number of rows in matrix \( A \).

This means, if matrix \( A \) is \( m \times n \), then matrix \( B \) must be \( n \times m \). In this case, both products \( AB \) and \( BA \) can be computed.

Why does this matter to know?

• It shows when double multiplication is possible, which can be helpful in more advanced topics like invertible matrices and determinants.
• It allows for full circular multiplication, where both orderings are possible.

Checking for these conditions can save the frustration of attempting impossible calculations.